Generalized Turán problem with bounded matching number

Abstract: For a graph $T$ and a set of graphs $\mathcal{H}$, let $\mbox{ex}(n,T,\mathcal{H})$ denote the maximum number of copies of $T$ in an $n$-vertex $\mathcal{H}$-free graph. Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value of $\mbox{ex}(n,K_2,{K_{k+1},M_{s+1}})$, where $K_{k+1}$ and $M_{s+1}$ are complete graph on $k+1$ vertices and matching of size $s+1$, respectively. Soon after, Gerbner~(arXiv2211.03272) continued the study by extending $K_{k+1}$ to general fixed graph $H$. In this paper, we continue the study of the function $\mbox{ex}(n, T,{H,M_{s+1}})$ when $T=K_r$ for $r\ge 3$. We determine the exact value of $\mbox{ex}(n,K_r,{K_{k+1},M_{s+1}})$ and give the value of $\mbox{ex}(n,K_r,{H,M_{s+1}})$ for general $H$ with an error term $O(1)$.